On K-derived quartics

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• 作者：Andrew Bremner ; ^{bremner@asu.edu" class="auth_mail" title="E-mail the corresponding author} ; Benjamin Carrillo ^{bcarril1@asu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11D25 ; secondary ; 11D41 ; 11G05 ; 11G30
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：168
• 期：Complete
• 页码：276-291
• 全文大小：313 K

文摘

Let K be a number field. A K  -derived polynomial f(x)∈K[x]$f(x)\in K[x]$ is a polynomial that factors into linear factors over K, as do all of its derivatives. Such a polynomial is said to be proper   if its roots are distinct. An unresolved question in the literature is whether or not there exists a proper Q$\mathbb{Q}$-derived polynomial of degree 4. Some examples are known of proper K-derived quartics for a quadratic number field K  , though other than $\mathbb{Q}(\sqrt{3})$, these fields have quite large discriminant. (The second known field is $\mathbb{Q}(\sqrt{3441})$.) The current paper describes a search for quadratic fields K over which there exist proper K  -derived quartics. The search finds examples for $K=\mathbb{Q}(\sqrt{D})$ with D=…,−95,−41,−39,−19,21,31,89,…$D=\dots ,-95,-41,-39,-19,21,31,89,\dots$ .